Question

The shortest distance between line $$y - x = 1$$ and curve $$x = {y^2}$$

A
34
B
328
C
832
D
43

Solution

The correct option is B $$\frac{{3\sqrt 2 }}{8}$$Let $$\left({a}^{2},a\right)$$ be the point of shortest distance on $$x={y}^{2}$$The distance between $$\left({a}^{2},a\right)$$ and line $$x−y+1=0$$ is given by$$D=\dfrac{{a}^{2}-a+1}{\sqrt{2}}$$$$=\dfrac{1}{\sqrt{2}}\left[{\left(a-\dfrac{1}{2}\right)}^{2}+\dfrac{3}{4}\right]$$It attains a minimum value at $$a=\dfrac{1}{2}$$$$\therefore\,{D}_{min}=\dfrac{1}{\sqrt{2}}\times \dfrac{3}{4}=\dfrac{3}{4\sqrt{2}}$$$$\therefore\,{D}_{min}=\dfrac{3}{4\sqrt{2}}\times\dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{3\sqrt{2}}{8}$$Mathematics

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